Sheng-Fang Huang

Solving an IVP by Laplace transformsThe Laplace transform method is a

powerful method for solving linear ODEs and corresponding initial value problems, as well as systems of ODEs arising in engineering.

6.1 Laplace Transform. Inverse Transform. Linearity. s-Shifting

ƒ(t) is a function defined for all t ≥ 0. Its Laplace transform, , is denoted by F(s), which is

(1)

Here we must assume that ƒ(t) is such that the integral exists (that is, has some finite value).

Inverse TransformThe given function ƒ(t) in (1) is called the

inverse transform of F(s) and is denoted by ; that is,

(1*)

Note that (1) and (1*) together imply = ƒ, and = F.

Example 1: Laplace TransformLet ƒ(t) = 1 when t ≥ 0. Find F(s).Solution. From (1) we obtain by

integration

The interval of integration in (1) is infinite. Such an integral is evaluated according to the rule

Example 2 Laplace Transform of the eat

Let ƒ(t) = eat when t ≥ 0, where a is a constant. FindSolution.

Linearity of the Laplace Transform

THEOREM 1

The Laplace transform is a linear operation; that is, for any functions ƒ(t) and g(t) whose transforms exist and any constants a and b the transform of aƒ(t) + bg(t) exists, and

Example 3 Application of Theorem 1: Hyperbolic FunctionsFind the transforms of cosh at and sinh at.Solution. Since coshat = 1/2(eat + e-at) and

sinhat = 1/2(eat – e-at),

Example 4 Cosine and SineDerive the formulas

Solution:

By substituting Ls into the formula for Lc on the right and then by substituting Lc into the formula for Ls on the right, we obtain

Solution by Transforms Using Derivatives. See next section.

Solution by Complex Methods. In Example 2, if we set a = iω with i = (–1)1/2, we obtain

Now by Theorem 1 and eiωt = cos ωt + i sin ωt

If we equate the real and imaginary parts of this and the previous equation, the result follows.

Some Functions ƒ(t) and Their Laplace Transforms

s-Shifting: Replacing s by s – a in the Transform

First Shifting Theorem, s-Shifting

THEOREM 2

If ƒ(t) has the transform F(s) (where s > k for some k), then eatƒ(t) has the transform F(s – a) (where s – a > k). In formulas,

or, if we take the inverse on both sides,

Example 5 s-Shifting: Damped Vibrations. Completing the Square

From Example 4,

For instance, use these formulas to find the inverse of the transform